Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach

In this paper we consider the time dependent Porous Medium Equation, uₜ = Δ u^γ with real polytropic exponent γ>1, subject to a homogeneous Dirichlet boundary condition. We are interested in recovering γ from the knowledge of the solution u at a given large time T. Based on an asymptotic inequality satisfied by the solution $u(T)$, we propose a numerical algorithm allowing us to recover γ. An upper bound for the error between the exact and recovered γ is then showed. Finally, numerical investigations are carried out in two dimensions.